The discussion revolves around finding the sum of a power series and determining its radius of convergence, specifically the series \(\sum_{n=1}^{\infty} (-1)^{n+1}\frac{(x-1)^n}{n}\). Participants are exploring the relationship between the series and its derivative.
Some participants have confirmed the form of the derivative of the function associated with the series. There is ongoing exploration of whether the resulting series can be classified as a geometric series, with prompts for further reasoning and clarification.
There is a hint provided regarding the derivative of the series, and participants are encouraged to consider the implications of their findings without reaching a definitive conclusion.
Bashyboy said:Homework Statement
Find the sum of the series and its radius of convergence:
[itex]\sum_{n=1}^{\infty} (-1)^{n+1}\frac{(x-1)^n}{n}[/itex]
Homework Equations
The Attempt at a Solution
I found the radius of convergence, but I wasn't sure how to find the sum of the power series.
Bashyboy said:Would it be [itex]f'(x) = \sum_{n=0}^{\infty} (-1)^{n+1}(x-1)^{n-1}[/itex]?
Bashyboy said:Would it be [itex]f'(x) = \sum_{n=0}^{\infty} (-1)^{n+1}(x-1)^{n-1}[/itex]?
Bashyboy said:If you distribute the (-1)^(n+1) to the (x-1)^(n-1) would it be a geometric series?